Theorem rabeqf

Description: Equality theorem for restricted class abstractions, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 7-Mar-2004)

	Ref	Expression
Hypotheses	rabeqf.1	|- F/_ x A
	rabeqf.2	|- F/_ x B
Assertion	rabeqf	|- ( A = B -> { x e. A | ph } = { x e. B | ph } )

Proof

Step	Hyp	Ref	Expression
1		rabeqf.1	|- F/_ x A
2		rabeqf.2	|- F/_ x B
3	1 2	nfeq	|- F/ x A = B
4		eleq2	|- ( A = B -> ( x e. A <-> x e. B ) )
5	4	anbi1d	|- ( A = B -> ( ( x e. A /\ ph ) <-> ( x e. B /\ ph ) ) )
6	3 5	abbid	|- ( A = B -> { x | ( x e. A /\ ph ) } = { x | ( x e. B /\ ph ) } )
7		df-rab	|- { x e. A | ph } = { x | ( x e. A /\ ph ) }
8		df-rab	|- { x e. B | ph } = { x | ( x e. B /\ ph ) }
9	6 7 8	3eqtr4g	|- ( A = B -> { x e. A | ph } = { x e. B | ph } )